A sharp coefficient bound and equality structure for quadratic representations of three-bit majority

A direct coefficient proof establishes a sharp bound for real quadratic majorants of three-bit majority on three labelled terminals and any finite set of original hidden bits, with affine terms free. The bound includes hidden-hidden and terminal-terminal premiums and even-boundary excess. Every original total-mass-six representation is characterized by four corner-star types with positive weights totaling one half per type, plus nonnegative unary isolates after hidden complements. Sharp constructions, a scalar ternary cubic formulation and cut/cone dictionaries are included. The full bound and equality family also follow from earlier parity-gadget results through explicit supplied deductions. Results retain candidate assurance status; historical novelty is unresolved and no priority claim is made.